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Expansion of q^(-1) * f(q) * f(q^7) / (f(-q^4) * f(-q^28)) in powers of q where f() is a Ramanujan theta function.
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%I #19 Mar 12 2021 22:24:47

%S 1,1,-1,0,1,0,-1,0,3,0,-2,0,2,0,-5,0,6,0,-7,0,7,0,-9,0,12,0,-13,0,16,

%T 0,-20,0,25,0,-27,0,31,0,-38,0,44,0,-51,0,58,0,-69,0,80,0,-92,0,102,0,

%U -118,0,141,0,-157,0,177,0,-203,0,234,0,-261,0,292,0

%N Expansion of q^(-1) * f(q) * f(q^7) / (f(-q^4) * f(-q^28)) in powers of q where f() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A230446/b230446.txt">Table of n, a(n) for n = -1..2500</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q^(-1) * chi(q) * chi(-q^2) * chi(q^7) * chi(-q^14) in power of q where chi() is a Ramanujan theta function.

%F Expansion of (eta(q^2) * eta(q^14))^3 / (eta(q) * eta(q^4)^2 * eta(q^7) * eta(q^28)^2) in powers of q.

%F Euler transform of period 28 sequence [ 1, -2, 1, 0, 1, -2, 2, 0, 1, -2, 1, 0, 1, -4, 1, 0, 1, -2, 1, 0, 2, -2, 1, 0, 1, -2, 1, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v - 2)^2 - u * v * (u - 2).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (28 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A123862.

%F a(n) = A161970(n) unless n=0. a(n) = -(-1)^n * A161970(n). a(2*n) = 0 unless n=0.

%e G.f. = 1/q + 1 - q + q^3 - q^5 + 3*q^7 - 2*q^9 + 2*q^11 - 5*q^13 + ...

%t a[ n_] := SeriesCoefficient[ q^-1 QPochhammer[ -q] QPochhammer[ -q^7] / (QPochhammer[ q^4] QPochhammer[ q^28]), {q, 0, n}]

%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^14 + A))^3 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^7 + A) * eta(x^28 + A)^2), n))}

%Y Cf. A123862, A161970.

%K sign

%O -1,9

%A _Michael Somos_, Oct 18 2013